Do you know what it means for a time series to be nonstationary?
Well, I would frankly need to review the ins and outs of that. But I’m running percentage returns, not prices (and not even first differences), so I wasn’t worried about unit roots. If you’re interested, here’s the Dickey-Fuller test:
. dfuller dailypret500, regress
Dickey-Fuller test for unit root Number of obs = 15473
---------- Interpolated Dickey-Fuller ---------
Test 1% Critical 5% Critical 10% Critical
Statistic Value Value Value
------------------------------------------------------------------------------
Z(t) -120.229 -3.430 -2.860 -2.570
------------------------------------------------------------------------------
MacKinnon approximate p-value for Z(t) = 0.0000
------------------------------------------------------------------------------
D. |
dailypret500 | Coef. Std. Err. t P>|t| [95% Conf. Interval]
-------------+----------------------------------------------------------------
dailypret500 |
L1 | -.9660763 .0080353 -120.23 0.000 -.9818264 -.9503262
_cons | .0003187 .0000777 4.10 0.000 .0001663 .000471
------------------------------------------------------------------------------
Unit roots are rejected at the 1% level.
…strictly speaking some financial analysts like to use log changes, but I doubt that presentation would make a big difference. I could be wrong though.
ETA: IIRC correctly, stationarity implies constant variance as well, which is something that financial returns don’t have. Variance tends to be auto-correlated.
I must admit that this characterization doesn’t exactly leap out from my sample. Here’s another try: “Data will reflect some sort of underlying structure. Errors may do the same, except that unobserved and even unobservable variables may play a prominent role. So absent evidence to the contrary, the distribution posited for the error term might be thought of as an approximation, possibly a rough one.”
Poking around the internet it seems that Easyfit is one of a few pieces of specialty software used for fitting lots of different distributions to a dataset. It seems that this sort of procedure isn’t a standard part of the usual general purpose statistical packages. Stata 8 fits the normal for example, but that’s all AFAIK. Then again, I know little about these tests: I don’t know whether measuring and matching moments would be straightforward.
Yes, you do, because your entire argument seems to be based on the assumption that financial models assume a stationary return distribution, which is demonstrably false.
You also don’t really need a test to see that the S&P 500 returns are not stationary. I did a quick plot of the returns over the period 1999-2010, and it’s immediately obvious that you’re not looking at a stationary series. Any longer time period would show much more variability.
Oh, but I agree financial returns can be and are modeled with GARCH: 2nd moments are autocorrelated after all. In fact I had GARCH in mind when I spoke of “Underlying structure”. But returns are also commonly modeled assuming normally distributed returns. Look at the Value at Risk literature. Consider the Black-Scholes model of options pricing. Read the business press when they speak of once every 10,000 year events that for some reason seem to recur every 5 years. Taleb wrote an entire book on this, which admittedly I haven’t read.
Again, I’m not claiming that financial professionals are unaware of these issues, although I guess I am saying that they are known to be blown off now and then. Permit me to quote from John Hull’s Options, Futures and Other Derivatives, 4th ed. He has a chapter on “Estimating Volatilities and Correlations” [with GARCH]:
Emphasis added. I’ll also note that Hull’s Fundamentals book lacks a chapter or mention of GARCH, although it is used to train stock traders.
Yup, my immediate thought was radioactive decay (since the number of particles is so huge (thank you, Avogadro) as to make it a nearly perfect normal distribution)
Also, the phsyics department at my Uni back in the day had a really neat demo lab. One of the demos was a impressively large Bean Machine. I used to love playing with that. And it always produced a nice looking normal curve. Not quite nature, since it was constructed by humans, but the principles are correct.
I believe that radioactive decay is a Poisson Process.
There are plenty of harmonic oscillators in nature for which this is a pretty good approximation – it works extremely well for diatomic molecules. If you use the better-fitting Morse potentialinstead of a perfect harmonic potential, the lowest order state is pretty close to a perfect Gaussian.
And, of course, the cross section through a TEM[sub]00[/sub] mode in a laser is a perfect Gaussian. In the real world, of course, there will invariably be dust specks in the beam, and the mirrors will be of finite extent. I doubt if anything in the real world can ever be a perfect Gaussian, because I doubt if anything in the real world will ever be a perfect function of any sort.
Ignorance fought Cal. Thanks to all the participants in this discussion.
I’d like to wind up a few loose threads.
I retract my speculation and will simply say that I don’t know the implications of non-normality on hypothesis testing. One advocate of robust regression [sup]1[/sup] states:
A less efficient estimator implies that we are not making the best guess possible of the parameter’s true value. The author goes on to say that informal eyeballing techniques can drop the efficiency loss to 10-20%: I guess he has outliers in mind. I’m not saying this guy is the final authority though: I’m just retracting my overly hasty remarks.
GARCH is a method of taking into account serially correlated variances. So in the context of the stock market, a large move today implies a large move tomorrow – though we won’t know the direction of that move. According to one set of authors[sup]2[/sup] while volatility clustering in normal GARCH models will increase the kurtosis of the series, it generally doesn’t do so sufficiently to reflect the kurtosis (or long tails) of financial market returns. Like other researchers, they opt for a GARCH model with non-Gauss innovations.
I would think that a normal GARCH model would produce zero skew, though I haven’t verified this. Eriksson and Forsberg (2004)[sup]3[/sup] use a GARCH model with conditional skewness. That seems to me to be an odd way of modelling momentum in returns, but I frankly don’t understand this properly. Anyway, they appear to use the Wald distribution in their GARCH model rather than Gauss.
Still, the OP is about the applicability and robustness of Gauss in general, and is not confined to financial markets. As computing power is cheap, it might not be a bad idea for the researcher to examine the descriptive statistics for empirical errors of uncertain provenance. But whether such a procedure would involve a risk of inappropriate data mining is something that I would have to think harder about.
ETA: Bean machines. No data. Discrete output. If it was made continuous by measuring impact location on a plate it would have an odd shape unless the bottom pins were moving.
[sup]1[/sup]See ROBUST INFERENCE by Frank Hampel (2000)
[sup]2[/sup]See Kurtosis of GARCH and Stochastic Volatility Models with Non-normal Innovations by Xuezheng Bai, Jeffrey R. Russell, George C. Tiao, July 27, 2001
[sup]3[/sup]See The Mean Variance Mixing GARCH (1,1) model -a new approach to estimate conditional skewness by Anders Eriksson Lars Forsberg 2004. All these working papers are available as .pdfs via google.
A bean machine meets the conditions of the Central Limit Theorem, so it’ll be a good approximation to within the limitations imposed by the binning, the finite number of beans, and the truncated tails. But of course you still have those limitations.
If man’s current understanding of physics is correct, then I would guess that a true normal distribution in nature is flat out impossible.
One property of normal distributions is that there is a finite probability of exceeding any value.
Thus, if the distribution of velocities of a set of particles is truly normal, then there is a finite chance that one or more of the particles will exceed the speed of light. Which is impossible if man’s current understanding of physics is correct.
Similarly, if the position of a particle after time t is normally distributed, then there is a finite chance that the particle moved faster than the speed of light.
Trivial Hijack!
The provenance of this quote has been less than clear for a few decades, but google resolves the matter. Poincare quotes a conversation with Gabriel Lippmann in Calcul des probabilités (1912). Others had difficulty locating the precise citation.
Boring Historical Details
Here’s the original french:
Emphasis in original: p. 170-71. Here is one translation combining google translate, my dismal high school french and some poetic license:
This doesn’t teach us much if we don’t have data on phi and psi. So we form an hypothesis for phi and call it The Law of Errors.
That can’t be deduced rigorously, though there are demonstrations such as the probability of the deviations is proportional to the differences.[sup]1[/sup] Everyone believes however, as Gabriel Lippmann told me one day, the experimenters because they imagine that it is a mathematical theorem and the mathematicians because it is an experimental fact.
Here’s how Gauss did it.
When we seek the best value to give to z, we have no alternative but to take the average of x1, x2, …, xn in the absence of any considerations that would justify a choice. Therefore the law of errors fits this mode of operation. Gauss looks for what should be the most likely value of phi, the mean value.
And here’s my paraphrase: as the great physicist Gabriel Lippmann once told Poincare, “Everybody believes in the Gaussian Law of Errors, the experimenters because they imagine that it is a mathematical theorem and the mathematicians because they believe it is an empirical fact.”
[sup]1[/sup]Translators note: huh?
The American astronomer and mathematician Charles Sanders Peirce wondered about the Law of Errors as well. To test it, he hired a laborer with no scientific background to respond to a signal by pressing a telegraph key. Peirce would measure the gap between the signal and the response in milliseconds. He did this for 24 days, 500 times per day. In 1872.
By the central limit theorem, he hypothesized that the resulting distribution would be Gauss. He was happy with his results: these graphs do indeed appear to be approximately normal[sup]1[/sup], as he came to label that distribution. This seemed to him to justify the use of least squares methods.
Sort of. His data was re-evaluated in 1928 by Edwin B. Wilson and Margartz M. Hilrirty. They concluded that the sample had many more outliers than a Gaussian and a positive skew as well. The dataset was revisited in 2009 by Roger Koenker of the University of Illinois using modern significance tests. Gaussian skewness was rejected in 19 out of 24 days; Gaussian kurtosis was rejected on all days. The author suggested that median approaches might be superior to mean ones, and that quantile approaches might be even better.
How did Peirce, sometimes referred to as one of the two greatest American scientists of the 1800s, form his conclusion? Well the plots actually do reveal some visual skew and kurtosis. But they also conform to Tukey’s Maxim: “All distributions are normal in the middle.”
[sup]1[/sup]See Peirce, C. S. (1873): On the Theory of Errors of Observation," Report of the Superintendent of the U.S. Coast Survey , pp. 200-224., Reprinted in The New Elements of Mathematics , (1976) collected papers of C.S. Peirce, ed. by C. Eisele, Humanities Press: Atlantic Highlands, N.J., vol. 3, part 1, 639-676.